Covering Dimension of C*-Algebras and 2-Coloured Classification

Peer reviewed

High-dimensional Z-stable AH algebras

Peer reviewed

Toeplitz operators on finite and infinite dimensional spaces with associated psi *-Fréchet algebras

The present thesis is a contribution to the multi-variable theory of Bergman and Hardy Toeplitz operators on spaces of holomorphic functions over finite and infinite dimensional domains. In particular, we focus on certain spectral invariant Frechet operator algebras F closely related to the local symbol behavior of Toeplitz operators in F. We summarize results due to B. Gramsch et.al. on the construction of Psi_0- and Psi^*-algebras in operator algebras and corresponding scales of generalized Sobolev spaces using commutator methods, generalized Laplacians and strongly continuous group actions. In the case of the Segal-Bargmann space H^2(C^n,m) of Gaussian square integrable entire functions on C^n we determine a class of vector-fields Y(C^n) supported in complex cones K. Further, we require that for any finite subset V of Y(C^n) the Toeplitz projection P is a smooth element in the Psi_0-algebra constructed by commutator methods with respect to V. As a result we obtain Psi_0- and Psi^*-operator algebras F localized in cones K. It is an immediate consequence that F contains all Toeplitz operators T_f with a symbol f of certain regularity in an open neighborhood of K. There is a natural unitary group action on H^2(C^n,m) which is induced by weighted shifts and unitary groups on C^n. We examine the corresponding Psi^*-algebra A of smooth elements in Toeplitz-C^*-algebras. Among other results sufficient conditions on the symbol f for T_f to belong to A are given in terms of estimates on its Berezin-transform. Local aspects of the Szegö projection P_s on the Heisenbeg group and the corresponding Toeplitz operators T_f with symbol f are studied. In this connection we apply a result due to Nagel and Stein which states that for any strictly pseudo-convex domain U the projection P_s is a pseudodifferential operator of exotic type (1/2, 1/2). The second part of this thesis is devoted to the infinite dimensional theory of Bergman and Hardy spaces and the corresponding Toeplitz operators. We give a new proof of a result observed by Boland and Waelbroeck. Namely, that the space of all holomorphic functions H(U) on an open subset U of a DFN-space (dual Frechet nuclear space) is a FN-space (Frechet nuclear space) equipped with the compact open topology. Using the nuclearity of H(U) we obtain Cauchy-Weil-type integral formulas for closed subalgebras A in H_b(U), the space of all bounded holomorphic functions on U, where A separates points. Further, we prove the existence of Hardy spaces of holomorphic functions on U corresponding to the abstract Shilov boundary S_A of A and with respect to a suitable boundary measure on S_A. Finally, for a domain U in a DFN-space or a polish spaces we consider the symmetrizations m_s of measures m on U by suitable representations of a group G in the group of homeomorphisms on U. In particular,in the case where m leads to Bergman spaces of holomorphic functions on U, the group G is compact and the representation is continuous we show that m_s defines a Bergman space of holomorphic functions on U as well. This leads to unitary group representations of G on L^p- and Bergman spaces inducing operator algebras of smooth elements related to the symmetries of U.

Quotients, exactness, and nuclearity in the operator system category

We continue our study of tensor products in the operator system category. We define operator system quotients and exactness in this setting and refine the notion of nuclearity by studying operator systems that preserve various pairs of tensor products. One of our main goals is to relate these refinements of nuclearity to the Kirchberg conjecture. In particular, we prove that the Kirchberg conjecture is equivalent to the statement that every operator system that is (min,er)-nuclear is also (el,c)-nuclear. We show that operator system quotients are not always equal to the corresponding operator space quotients and then study exactness of various operator system tensor products for the operator system quotient. We prove that an operator system is exact for the min tensor product if and only if it is (min,el)-nuclear. We give many characterizations of operator systems that are (min,er)-nuclear, (el,c)-nuclear, (min,el)-nuclear and (el,max)-nuclear. These characterizations involve operator system analogues of various properties from the theory of C*-algebras and operator spaces, including the WEP and LLP.

The C*-algebra of ax+b-like groups

We describe the C*-algebras of " ax+b" -like groups in terms of algebras of operator fields defined over their dual spaces.

The solvable Lie group N_{6, 28}: an example of an almost C_0(\mathcal{K})-C*-algebra

Motivated by the description of the C*-algebra of the affine automorphism group N6,28 of the Siegel upper half-plane of degree 2 as an algebra of operator fields defined over the unitary dual View the MathML source of the group, we introduce a family of C*-algebras, which we call almost C0(K), and we show that the C*-algebra of the group N6,28 belongs to this class.

INTEGRAL FUNCTIONALS ON C*-ALGEBRA OF VECTOR-VALUED REGULATED FUNCTIONS

In this paper we deal with the notion of regulated functions with values in a C*-algebra A and present examples using a special bi-dimensional C*-algebra of triangular matrices. We consider the Dushnik integral for these functions and shows that a convenient choice of the integrator function produces an integral homomorphism on the C*-algebra of all regulated functions ([a, b], A). Finally we construct a family of linear integral functionals on this C*-algebra.

Pseudodifferential analysis in Psi*-algebras on transmission spaces, infinite solving ideal chains and K-theory for conformally compact spaces

The present thesis is a contribution to the theory of algebras of pseudodifferential operators on singular settings. In particular, we focus on the $b$-calculus and the calculus on conformally compact spaces in the sense of Mazzeo and Melrose in connection with the notion of spectral invariant transmission operator algebras. We summarize results given by Gramsch et. al. on the construction of $Psi_0$-and $Psi*$-algebras and the corresponding scales of generalized Sobolev spaces using commutators of certain closed operators and derivations. In the case of a manifold with corners $Z$ we construct a $Psi*$-completion $A_b(Z,{}^bOmega^{1/2})$ of the algebra of zero order $b$-pseudodifferential operators $Psi_{b,cl}(Z, {}^bOmega^{1/2})$ in the corresponding $C*$-closure $B(Z,{}^bOmega^{12})hookrightarrow L(L^2(Z,{}^bOmega^{1/2}))$. The construction will also provide that localised to the (smooth) interior of Z the operators in the $A_b(Z, {}^bOmega^{1/2})$ can be represented as ordinary pseudodifferential operators. In connection with the notion of solvable $C*$-algebras - introduced by Dynin - we calculate the length of the $C*$-closure of $Psi_{b,cl}^0(F,{}^bOmega^{1/2},R^{E(F)})$ in $B(F,{}^bOmega^{1/2}),R^{E(F)})$ by localizing $B(Z, {}^bOmega^{1/2})$ along the boundary face $F$ using the (extended) indical familiy $I^B_{FZ}$. Moreover, we discuss how one can localise a certain solving ideal chain of $B(Z, {}^bOmega^{1/2})$ in neighbourhoods $U_p$ of arbitrary points $pin Z$. This localisation process will recover the singular structure of $U_p$; further, the induced length function $l_p$ is shown to be upper semi-continuous. We give construction methods for $Psi*$- and $C*$-algebras admitting only infinite long solving ideal chains. These algebras will first be realized as unconnected direct sums of (solvable) $C*$-algebras and then refined such that the resulting algebras have arcwise connected spaces of one dimensional representations. In addition, we recall the notion of transmission algebras on manifolds with corners $(Z_i)_{iin N}$ following an idea of Ali Mehmeti, Gramsch et. al. Thereby, we connect the underlying $C^infty$-function spaces using point evaluations in the smooth parts of the $Z_i$ and use generalized Laplacians to generate an appropriate scale of Sobolev spaces. Moreover, it is possible to associate generalized (solving) ideal chains to these algebras, such that to every $ninN$ there exists an ideal chain of length $n$ within the algebra. Finally, we discuss the $K$-theory for algebras of pseudodifferential operators on conformally compact manifolds $X$ and give an index theorem for these operators. In addition, we prove that the Dirac-operator associated to the metric of a conformally compact manifold $X$ is not a Fredholm operator.

Upregulation of c-Myc may contribute to the pathogenesis of canine pemphigus vulgaris

The pathomechanism in human pemphigus vulgaris (PV) has recently been described to rely on generalized c-Myc upregulation in skin and oral mucosa followed by hyperproliferation. Here we assessed whether dogs suffering from PV present the same pathological changes as described for human patients with PV. Using immunofluorescence analysis on patients' biopsy samples, we observed marked nuclear c-Myc accumulation in all layers of the epidermis and oral mucosa in all (3/3) dogs analysed. In addition, c-Myc upregulation was accompanied by an increased number of proliferating Ki67-positive cells. These molecular changes were further paralleled by deregulated expression of wound healing and terminal differentiation markers as observed in human PV. Together these findings suggest a common pathomechanism for both species which is of particular relevance in the light of the recently discussed novel therapeutic strategies aiming at targeting PV antibody-induced signalling cascades.

Orientation in operator algebras

A concept of orientation is relevant for the passage from Jordan structure to associative structure in operator algebras. The research reported in this paper bridges the approach of Connes for von Neumann algebras and ourselves for C*-algebras in a general theory of orientation that is of geometric nature and is related to dynamics.

Integrin regulation of c-Abl tyrosine kinase activity and cytoplasmic–nuclear transport

The product of the c-abl protooncogene is a nonreceptor tyrosine kinase found in both the cytoplasm and the nucleus. We report herein that cell adhesion regulates the kinase activity and subcellular localization of c-Abl. When fibroblastic cells are detached from the extracellular matrix, kinase activity of both cytoplasmic and nuclear c-Abl decreases, but there is no detectable alteration in the subcellular distribution. Upon adhesion to the extracellular matrix protein fibronectin, a transient recruitment of a subset of c-Abl to early focal contacts is observed coincident with the export of c-Abl from the nucleus to the cytoplasm. The cytoplasmic pool of c-Abl is reactivated within 5 min of adhesion, but the nuclear c-Abl is reactivated after 30 min, correlating closely with its return to the nucleus and suggesting that the active nuclear c-Abl originates in the cytoplasm. In quiescent cells where nuclear c-Abl activity is low, the cytoplasmic c-Abl is similarly regulated by adhesion but the nuclear c-Abl is not activated upon cell attachment. These results show that c-Abl activation requires cell adhesion and that this tyrosine kinase can transmit integrin signals to the nucleus where it may function to integrate adhesion and cell cycle signals.

Functional interaction between c-Jun and promoter factor Sp1 in epidermal growth factor-induced gene expression of human 12(S)-lipoxygenase

The functional role of the interaction between c-Jun and simian virus 40 promoter factor 1 (Sp1) in epidermal growth factor (EGF)-induced expression of 12(S)-lipoxygenase gene in human epidermoid carcinoma A431 cells was studied. Coimmunoprecipitation experiments indicated that EGF stimulated interaction between c-Jun and Sp1 in a time-dependent manner. Overexpression of Ha-ras and c-Jun also enhanced the amount of c-Jun binding to Sp1. In addition, the c-Jun dominant negative mutant TAM-67 not only inhibited the coimmunoprecipitated c-Jun binding to Sp1 in a dose-dependent manner in cells overexpressing c-Jun but also reduced promoter activity of the 12(S)-lipoxygenase gene induced by c-Jun overexpression. Treatment of cells with EGF increased the interaction between the Sp1 oligonucleotide and nuclear c-Jun/Sp1 in a time-dependent manner. Furthermore, EGF activated the chimeric promoter consisting of 10 tandem GAL4-binding sites, which replaced the three Sp1-binding sites in the 12(S)lipoxygenase promoter only when coexpressed with GAL4-c-Jun () fusion proteins. These results indicate that the direct interaction between c-Jun and Sp1 induced by EGF cooperatively activated expression of the 12(S)-lipoxygenase gene, and that Sp1 may serve at least in part as a carrier bringing c-Jun to the promoter, thus transactivating the transcriptional activity of 12(S)-lipoxygenase gene.

The stress response to ionizing radiation involoves c-Abl-dependent phosphorylation of SHPTP1.

c-Abl is a nonreceptor tyrosine kinase that is activated by certain DNA-damaging agents. The present studies demonstrate that nuclear c-Abl binds constitutively to the protein tyrosine phosphatase SHPTP1. Treatment with ionizing radiation is associated with c-Abl-dependent tyrosine phosphorylation of SHPTP1. The results demonstrate that the SH3 domain of c-Abl interacts with a WPDHGVPSEP motif (residues 417-426) in the catalytic domain of SHPTP1 and that c-Abl phosphorylates C terminal Y536 and Y564 sites. The functional significance of the c-Abl-SHPTP1 interaction is supported by the demonstration that, like c-Abl, SHPTP1 regulates the induction of Jun kinase activity following DNA damage. These findings indicate that SHPTP1 is involved in the response to genotoxic stress through a c-Abl-dependent mechanism.

Spectral synthesis for Banach Algebras II

This paper continues the study of spectral synthesis and the topologies tau-infinity and tau-r on the ideal space of a Banach algebra, concentrating particularly on the class of Haagerup tensor products of C*-algebras. For this class, it is shown that spectral synthesis is equivalent to the Hausdorffness of tau_infinity. Under a weak extra condition, spectral synthesis is shown to be equivalent to the Hausdorffness of tau_r.

Reconstruction of graded groupoids from graded Steinberg algebras

We show how to reconstruct a graded ample Hausdorff groupoid with topologically principal neutrally-graded component from the ring structure of its graded Steinberg algebra over any commutative integral domain with 1, together with the embedding of the canonical abelian subring of functions supported on the unit space. We deduce that diagonal-preserving ring isomorphism of Leavitt path algebras implies $C^*$-isomorphism of $C^*$-algebras for graphs $E$ and $F$ in which every cycle has an exit. This is a joint work with Joan Bosa, Roozbeh Hazrat and Aidan Sims.